In tae-kwon-do, a hand is slammed down onto a target at a speed of 13 m/s and comes to a stop during the 5.5 ms collision. Assume that during the impact the hand is independent of the arm and has a mass of 0.70 kg. What are the magnitudes of the (a) impulse and (b) average force on the hand from the target?

Answer

To solve the problem, we need to calculate the impulse and the average force exerted on the hand during the collision with the target. Let's break this down step by step.

### Step 1: Calculate the initial momentum of the hand.
The initial momentum (p_initial) can be calculated using the formula:
\[ p_{\text{initial}} = m \cdot v \]
where:
- \( m = 0.70 \, \text{kg} \) (mass of the hand)
- \( v = 13 \, \text{m/s} \) (initial velocity of the hand)

Calculating this gives:
\[ p_{\text{initial}} = 0.70 \, \text{kg} \cdot 13 \, \text{m/s} = 9.1 \, \text{kg m/s} \]

### Step 2: Calculate the final momentum of the hand.
Since the hand comes to a stop after hitting the target, the final velocity (v_final) is 0 m/s. Thus, the final momentum (p_final) is:
\[ p_{\text{final}} = m \cdot v_{\text{final}} = 0.70 \, \text{kg} \cdot 0 \, \text{m/s} = 0 \, \text{kg m/s} \]

### Step 3: Calculate the change in momentum (impulse).
Impulse (J) is defined as the change in momentum:
\[ J = p_{\text{final}} - p_{\text{initial}} \]
Substituting the values we found:
\[ J = 0 - 9.1 \, \text{kg m/s} = -9.1 \, \text{kg m/s} \]
Since impulse is a magnitude, we take the absolute value:
\[ |J| = 9.1 \, \text{N s} \]

### Step 4: Calculate the average force exerted on the hand.
The average force (F) can be calculated using the formula:
\[ F = \frac{J}{\Delta t} \]
where \( \Delta t \) is the time duration of the collision. Given \( \Delta t = 5.5 \, \text{ms} = 5.5 \times 10^{-3} \, \text{s} \):
\[ F = \frac{9.1 \, \text{N s}}{5.5 \times 10^{-3} \, \text{s}} \]

Calculating this gives:
\[ F = \frac{9.1}{5.5 \times 10^{-3}} \approx 1654.55 \, \text{N} \]
Rounding this gives:
\[ F \approx 1.7 \times 10^{3} \, \text{N} \]

### Final Answers:
(a) Impulse = 9.1 N s
(b) Average Force = 1.7 × 10³ N

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